
The second one is McDiarmid's inequality [1]}.
[]mylemmaMcdiarmid

Let \(X_{1},...,X_{n} \in \mathbb {M}\)  be independent random variables, and \(f\)  be a mapping \(f:\mathbb {M}^{n} \rightarrow \mathbb {R}\) . If, for all \(i \in \lbrace 1,...,n\rbrace \) , and for all \(x_{1},...,x_{n},x_{i^{^{\prime }}} \in \mathbb {M}\) , the function \(f\)  satisfies
\(|f(x_{1},...,x_{i-1},x_{i},x_{i+1},...,x_{n})-f(x_{1},...,x_{i-1},x_{i^{^{\prime }}},x_{i+1},...,x_{n})| \le c_{i}~,\) 
